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Inverse of a matrix using elementary row operations

Many a times, we need to find inverse of a given matrix. Let us look at one of the easiest ways to find the inverse of an invertible matrix.

6 minutes long
Posted by Tanwir Silar, 4/8/2021
Inverse of a matrix using elementary row operations

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The need to find the inverse of a matrix in matrix algebra is immense. This brings us to one of the easiest ways to find the inverse of a matrix – using elementary row operations.

What are elementary operations of a matrix?

  1. Interchange of any two rows or two columns. Symbolically written as, C_i bidirectional C_j and R_j bidirectional R_j
  2. The multiplication of the elements of any row or column by a non-zero number. Symbolically shown as, r_i changes to k times r_j and c_i changes to k times c_j
  3. The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number. Represented as, and

Invertible Matrices

A square matrix A is said to invertible if its determinant has a non-zero value, i.e., A is non-singular.
For a square matrix A of order n, its inverse is denoted by Matrix A inverse in bold and has an order n. The product of a matrix and its inverse results to the Identity matrix of order n.Multiplication of a matrix with its inverse We can observe that, if B is the inverse of A, then A is the inverse of B.
The notion of inverse can be used to calculate an unknown matrix within multiplication.


Example:
To know the values of X, we need to multiply both sides of the equation with the inverse of A. Here, the order of X, A, B is same.
Inverse of a matrix and Matrix multiplication
If A and B are known, then X can be calculated using above steps.

Elementary row operations – Inverse of matrix

Before diving into calculating the inverse of a matrix, let us know more about what operations should be applied to the product AB, if elementary operations (transformations) are performed on X, so that the equation X = AB holds. Row operations carried out on X should also be performed on the first matrix, A, of the product AB. Similarly, if column operations were carried out on X then they should also be carried on the second matrix, B, of the product B.

Procedure
1. We write Multiplicative identity, and then apply a sequence of row operations on A till we get, calculating the inverse of a matrix using elementary operations.

Example:
Using elementary operations, calculate the inverse of the matrix


Matrix A for the example

Solution:
We start by writing Multiplicative identity.
part 1 of the solution for example
part 2 of the solution for example
We have been able to write Multiplicative identity as calculating the inverse of a matrix using elementary operations where B is the inverse of matrix A.
Hence, the inverse is

final answer for the inverse example

We can validate the answer by multiplying it with A, as Multiplication of a matrix with its inverse.



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